Question: What is the average rate of change of the function $f(x)=2^x$ over the interval $[3,3+h]$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\dfrac{2^{ h}-2^{ 3}}{h}$ (Choice B) B $\dfrac{2^{{3+h}}-2^{3}}{h}$ (Choice C) C $\dfrac{2^{{3+h}}-2^{3}}{3}$ (Choice D) D $\dfrac{2^{{3+h}}-2^{3}}{3+h}$
Solution: This is the formula for the average rate of change of a function $f$ over the interval $[a,b]$ : $\dfrac{f(b)-f(a)}{b-a}$ We are interested in the average rate of change of $f(x)=2^x$ over the interval $[3,3+h]$ : $\begin{aligned} &\phantom{=}\dfrac{f(3+h)-f(3)}{(3+h)-(3)} \\\\ &=\dfrac{2^{{3+h}}-2^{3}}{3+h-3} \\\\ &=\dfrac{2^{{3+h}}-2^{3}}{h} \end{aligned}$ The average rate of change of the function is $\dfrac{2^{{3+h}}-2^{3}}{h}$. Notice that the average rate of change is calculated just like the slope of the secant line that intersects the graph of the function at the interval's endpoints.